Difference between the two quaternions

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I\'m making a 3D portal system in my engine (like Portal game). Each of the portals has its own orientation saved in a quaternion. To render the virtu

Answer 1

Quaternions work the following way: the local frame of reference is represented as the imaginary quaternion directions i,j,k. For instance, for an observer standing in the portal door 1 and looking in the direction of the arrow, direction i may represent the direction of the arrow, j is up and k=ij points to the right of the observer. In global coordinates represented by the quaternion q1, the axes in 3D coordinates are

q1*(i,j,k)*q1^-1=q1*(i,j,k)*q1',

where q' is the conjugate, and for unit quaternions, the conjugate is the inverse.

Now the task is to find a unit quaternion q so that directions q*(i,j,k)*q' in local frame 1 expressed in global coordinates coincide with the rotated directions of frame 2 in global coordinates. From the sketch that means forwards becomes backwards and left becomes right, that is

q1*q*(i,j,k)*q'*q1'=q2*(-i,j,-k)*q2'
                   =q2*j*(i,j,k)*j'*q2'

which is readily achieved by equating

q1*q=q2*j or q=q1'*q2*j.

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But details may be different, mainly that another axis may represent the direction "up" instead of j.

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If the global system of the sketch is from the bottom, so that global-i points forward in the vertical direction, global-j up and global-k to the right, then local1-(i,j,k) is global-(-i,j,-k), giving

q1=j. 

local2-(i,j,k) is global-(-k,j,i) which can be realized by

q2=sqrt(0.5)*(1+j), 

since

(1+j)*i*(1-j)=i*(1-j)^2=-2*i*j=-2*k and 
(1+j)*k*(1-j)=(1+j)^2*k= 2*j*k= 2*i

Comparing this to the actual values in your implementation will indicate how the assignment of axes and quaternion directions has to be changed.